Finite Group Databasebeta

← #211106

Subgroups — #211106

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23716C211yesmaximal, characteristic
2#17891C211yesmaximal, characteristic, complemented
3#17869C211yesmaximal, characteristic, complemented
4C2×#1581C211yesmaximal, characteristic, complemented
5#17331C211yesmaximal, characteristic, complemented
6#9668C211yesmaximal, characteristic, complemented
7#9608C211yesmaximal, characteristic, complemented
8#9552C211yesmaximal, characteristic, complemented
9#9552C211yesmaximal, characteristic, complemented
10#9095C211yesmaximal, characteristic, complemented
11#9095C211yesmaximal, characteristic, complemented
12#8606C211yesmaximal, characteristic, complemented
13#8297C211yesmaximal, characteristic, complemented
14#8166C211yesmaximal, characteristic, complemented
15C8×#100C211yesmaximal, characteristic, complemented
16#2965C2211yescharacteristic
17C8×#41C2211yescharacteristic
18C4×AGL(1,ℤ/8)C2211yescharacteristic
19C4×AGL(1,ℤ/8)C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23#2922C2211yescharacteristic
24#2089C411yescharacteristic
25#2078C411yescharacteristic
26#2063C411yescharacteristic
27#2063C411yescharacteristic
28#2051C411yescharacteristic
29#2051C411yescharacteristic
30#2045C411yescharacteristic
31#2018C411yescharacteristic
32C4×#108C2211yescharacteristic
33C4×#100C2211yescharacteristic
34C4×#100C2211yescharacteristic
35#1594—21no—
36#1592—21no—
37#1590—21no—
38#1589—21no—
39#1586—21no—
40#1585—21no—
41#1582—21no—
42#1581—21no—
43#1581—21no—
44#1581—21no—
45#1581—21no—
46#1581—21no—
47#1565C2211yescharacteristic
48#1565C2211yescharacteristic
49#1564C2211yescharacteristic
50C2×#335C2211yescharacteristic
51#1478C2211yescharacteristic
52#1478C2211yescharacteristic
53#1477C2211yescharacteristic
54#1477C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1460C2211yescharacteristic
58#1460C2211yescharacteristic
59#1454—21no—
60#1453—21no—
61#1452—21no—
62#1451C2211yescharacteristic
63#1451C2211yescharacteristic
64#1450C2211yescharacteristic
65C8×M16—21no—
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1283C2211yescharacteristic
72#1283C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×AGL(1,ℤ/8)C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C411yescharacteristic
77C22×D8C2×C411yescharacteristic
78C22×M16—21no—
79C23×C8—21no—
80C4×#41C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83#446D411yescharacteristic
84#446D411yescharacteristic
85C8×Q8—21no—
86#442—21no—
87#442—21no—
88#440—21no—
89#440—21no—
90C4×SD16—21no—
91C4×SD16—21no—
92C4×D8—21no—
93C4×D8—21no—
94C4×D8—21no—
95C4×D8—21no—
96C4×D8—21no—
97C4×D8—21no—
98#436—21no—
99#436—21no—
100#436—21no—
101#436—21no—
102#436D411yescharacteristic
103#436D411yescharacteristic
104#435—21no—
105#435—21no—
106C8×D4—21no—
107C8×D4—22no—
108C8×D4—21no—
109C8×D4—21no—
110C8×D4—21no—
111C8×D4—21no—
112#433—21no—
113#433—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121#431C2311yescharacteristic
122C2×#108C2×C411yescharacteristic
123C2×#108C2×C411yescharacteristic
124#428C2×C411yescharacteristic
125#428C2×C411yescharacteristic
126C2×#107C2×C411yescharacteristic
127#424C2×C411yescharacteristic
128#424C2×C411yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131#418C2×C411yescharacteristic
132#418C2×C411yescharacteristic
133#417C2×C411yescharacteristic
134#417C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×#100C2×C411yescharacteristic
138C2×#100C2×C411yescharacteristic
139C2×#100C2×C411yescharacteristic
140C2×#100C2×C411yescharacteristic
141#408C2×C411yescharacteristic
142#408C2×C411yescharacteristic
143C2×C22⋊C8—21no—
144C2×C22⋊C8—21no—
145C4×M16—21no—
146C4×M16C2311yescharacteristic
147C4×M16C2311yescharacteristic
148C4×M16C2311yescharacteristic
149C2×C4×C8—21no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153#343C2×C411yescharacteristic
154#343C2×C411yescharacteristic
155#343C2×C411yescharacteristic
156#343C2×C411yescharacteristic
157#341C2×C411yescharacteristic
158#341C2×C411yescharacteristic
159#335—21no—
160#335—21no—
161#335—21no—
162#335—21no—
163#335—21no—
164#334—21no—
165#334—21no—
166#334—21no—
167Q8⋊C8—21no—
168Q8⋊C8—21no—
169D4⋊C8—21no—
170D4⋊C8—21no—
171D4⋊C8—21no—
172D4⋊C8—21no—
173D4⋊C8—21no—
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179#323—22no—
180#323—22no—
181#322—21no—
182#322—21no—
183#322—21no—
184#322—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322D411yescharacteristic
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192C82—21no—
193C82—21no—
194C82—21no—
195C2×#41C22×C411yescharacteristic
196C22×D4C22×C411yescharacteristic
197C22×D4C22×C411yescharacteristic
198C23×C4—21no—
199C23×C4—21no—
200AGL(1,ℤ/8)—22no—
201AGL(1,ℤ/8)—22no—
202AGL(1,ℤ/8)—41no—
203AGL(1,ℤ/8)—41no—
204C2×SD16—21no—
205C2×SD16—21no—
206C2×D8—21no—
207C2×D8—21no—
208C2×D8—22no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—41no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—41no—
215C2×M16—41no—
216C2×M16—82no—
217C2×M16C22×C411yescharacteristic
218C2×M16C22×C411yescharacteristic
219C2×M16C22×C411yescharacteristic
220C2×M16C22×C411yescharacteristic
221C2×M16C22×C411yescharacteristic
222C2×M16C22×C411yescharacteristic
223C22×C8—22no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—41no—
228C22×C8—41no—
229C22×C8—82no—
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8#4111yescharacteristic
233C22×C8#4111yescharacteristic
234C4×Q8C2×D411yescharacteristic
235C4×Q8#4111yescharacteristic
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4C2×D411yescharacteristic
247C4×D4#4111yescharacteristic
248#117—21no—
249#117C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C42—21no—
255C2×C42C2411yescharacteristic
256#108—22no—
257#108—41no—
258#107—21no—
259#107—21no—
260#107—21no—
261#107—21no—
262#106—21no—
263#106—21no—
264C4⋊C8—22no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8C2×D411yescharacteristic
279C4⋊C8C2×D411yescharacteristic
280#103—21no—
281#103—21no—
282#102—21no—
283#102—21no—
284#102—21no—
285#102—21no—
286#102—21no—
287#102—21no—
288#102—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292#100—22no—
293#100—22no—
294#100—41no—
295#100—41no—
296C22⋊C8—22no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—41no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309C22⋊C8—41no—
310C22⋊C8#4112yes—
311#97—21no—
312#97—21no—
313#97—21no—
314#97—21no—
315#97—21no—
316C4×C8—22no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8#4111yescharacteristic
334C4×C8#4111yescharacteristic
335C24—21no—
336C24—21no—
337#41—21no—
338#41—21no—
339#41—21no—
340#41—21no—
341C2×Q8C4×D411yescharacteristic
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4C4×D411yescharacteristic
355C2×D4#13112yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4C23×C411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372SD16—42no—
373SD16—42no—
374D8—42no—
375D8—42no—
376D8—42no—
377D8—42no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—21no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—42no—
390M16—42no—
391M16—82no—
392M16—82no—
393M16#13112yes—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—22no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8—21no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—82no—
416C2×C8—81no—
417C2×C8—82no—
418C2×C8—82no—
419C2×C8—81no—
420C2×C8—81no—
421C2×C8C4×D411yescharacteristic
422C2×C8C4×D411yescharacteristic
423C2×C8C4×D411yescharacteristic
424C2×C8C4×D411yescharacteristic
425C2×C8#12111yescharacteristic
426C2×C8#12111yescharacteristic
427C2×C8#12111yescharacteristic
428C2×C8#12111yescharacteristic
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4C4×D411yescharacteristic
436C4⋊C4C4×D411yescharacteristic
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42C22×D411yescharacteristic
448C42C22×D411yescharacteristic
449C42C2×#4111yescharacteristic
450C42C2×#4111yescharacteristic
451C23—21no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23C4×#4111yescharacteristic
463Q8—21no—
464Q8—21no—
465D4—22no—
466D4—21no—
467D4—21no—
468D4—22no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×#12111yescharacteristic
501C2×C4#52511yescharacteristic
502C2×C4#52911yescharacteristic
503C2×C4#53211yescharacteristic
504C2×C4C2×#13111yescharacteristic
505C2×C4C2×#13111yescharacteristic
506C2×C4#56811yescharacteristic
507C2×C4#56811yescharacteristic
508C8—21no—
509C8—21no—
510C8—22no—
511C8—21no—
512C8—21no—
513C8—21no—
514C8—21no—
515C8—21no—
516C8—42no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—81no—
527C8—81no—
528C8—81no—
529C8—81no—
530C8#47212yes—
531C22—21no—
532C22—21no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—82no—
538C22—82no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—161no—
545C22—161no—
546C22C4×#12111yescharacteristic
547C22#296611yescharacteristic
548C22#296711yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—81no—
556C4—81no—
557C4#293311yescharacteristic
558C4#293311yescharacteristic
559C4#293311yescharacteristic
560C4#293511yescharacteristic
561C4C2×#47211yescharacteristic
562C4#306711yescharacteristic
563C2—21no—
564C2—21no—
565C2—81no—
566C2—82no—
567C2—161no—
568C2#2149611yescharacteristic
569C2C4×#47211yescharacteristic
570C2#2155911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#2018C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #17891C2semidirect
3. #17869C2semidirect
4. C2×#1581C2semidirect
5. #17331C2semidirect
6. #9668C2semidirect
7. #9608C2semidirect
8. #9552C2semidirect
9. #9552C2semidirect
10. #9095C2semidirect
11. #9095C2semidirect
12. #8606C2semidirect
13. #8297C2semidirect
14. #8166C2semidirect
15. C8×#100C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237162
#178912
#178692
C2×#15812
#173312
#96682
#96082
#95522
#90952
#86062
#82972
#81662
C8×#1002